Priority queueing systems: from probability generating functions to tail probabilities Tom Maertens ∗ , Joris Walraevens, Herwig Bruneel Ghent University – UGent Department of Telecommunications and Information Processing SMACS Research Group Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium Phone: +32-9-2648901 Fax: +32-9-2644295 E-mail: {tmaerten,jw,hb}@telin.UGent.be Abstract Obtaining (tail) probabilities from a transform function is an important topic in queueing theory. To obtain these probabilitie s in discre te-time queueing systems, we have to invert probability genera ting functions, since most important distributions in discrete-time queueing systems can be determined in the form of probability generating functions. In this paper, we calculate the tail probabilities of two particular random variables in discrete-time priority queueing systems, by means of the dominant singularity ap- proximation. We show that obtaining these tail probabilities can be a complex task, and that the obtained tail probabilitie s are not neces saril y exponen tial (as in most ’tradi tional ’ queue ing systems). Furthe r, we show the impact and significance of the various system parameters on the type of tail behavior. Finally, we compare our approximation results with simulations. keywords priority queueing systems, tail probabilities, dominant pole, non-exponential behavior 1 In tr oduction Many probab ilit y distributio ns of interest in queue ing models can be deter mined in the form of trans- forms: the Laplace-Stieltjes transforms ofcontinuous density functions or the z -transforms ofdiscrete probability mass functions. The benefit of using transforms in analyses with stochastic variables has been frequently demonstrated in the past. Transforms are furthermore very useful to extract numerical results, e.g., to calculate moments. However, a seeming disadvantage of working with transforms is that it is not always easy to explicitly calculate the corresponding cumulative distribution functions (cdf’s), probability density functions (pdf), or probability mass functions (pmf’s). Often, we are only interested in the tail of the probability distribution. Tail probabilities typically represent the ’exceptional’ situations in a queueing system (or more generally, a communication network), of which we want to estimate the frequency of. E.g. the probability that the delay is larger than a given ∗ Corresponding author 1
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Obtaining (tail) probabilities from a transform function is an important topic in queueing theory. Toobtain these probabilities in discrete-time queueing systems, we have to invert probability generatingfunctions, since most important distributions in discrete-time queueing systems can be determined in theform of probability generating functions. In this paper, we calculate the tail probabilities of two particularrandom variables in discrete-time priority queueing systems, by means of the dominant singularity ap-proximation. We show that obtaining these tail probabilities can be a complex task, and that the obtainedtail probabilities are not necessarily exponential (as in most ’traditional’ queueing systems). Further, weshow the impact and significance of the various system parameters on the type of tail behavior. Finally,we compare our approximation results with simulations.
Many probability distributions of interest in queueing models can be determined in the form of trans-
forms: the Laplace-Stieltjes transforms of continuous density functions or the z-transforms of discrete
probability mass functions. The benefit of using transforms in analyses with stochastic variables has been
frequently demonstrated in the past. Transforms are furthermore very useful to extract numerical results,e.g., to calculate moments. However, a seeming disadvantage of working with transforms is that it is not
always easy to explicitly calculate the corresponding cumulative distribution functions (cdf’s), probability
density functions (pdf), or probability mass functions (pmf’s).
Often, we are only interested in the tail of the probability distribution. Tail probabilities typically
represent the ’exceptional’ situations in a queueing system (or more generally, a communication network),
of which we want to estimate the frequency of. E.g. the probability that the delay is larger than a given
value N or the packet loss are examples of interesting performance measures for which the calculation
of the (asymptotic behavior of) tail probabilities is usually sufficient. Obtaining tail probabilities of a
stochastic variable from its transform, which is basically an inversion problem, is thus an important
topic in queueing theory. A theoretical solution method is to analytically invert the transform, yielding an
explicit , closed-form expression for the underlying probability distribution. However, this is only possible if
the transform expressions are simple enough. In sophisticated queueing models, this method is practically
infeasible. Therefore, one has to look for approximate solutions.
From our point of view, existing approximate inversion techniques can be roughly divided into two
categories: numerical inversion methods and analytical inversion methods. Abate and Whitt [4] provide an
extensive study of various numerical methods for transform inversion. In general, theoretical solutions for
the inversion problem can usually be expressed via integrals (see e.g. [9] for a short review): a line integral
in the continuous case or a contour integral in the discrete case. These basic inversion integrals can then
be calculated by performing a numerical integration. The Fourier-series method numerically integrates a
standard inversion integral by means of the trapezoidal rule (see e.g. [4]). In [7] (Laplace transforms) and
[5] (probability generating functions), the authors propose Poisson summation formulas which identify the
discretization errors associated with this trapezoidal rule. Algorithms based on the Fourier-series method
are further a.o. presented in [19] and [21]. Most of these algorithms require the evaluation of the involved
transforms at many complex numbers. However, if the transform is only characterised implicitly via a
functional equation (e.g., the busy-period distribution in a GI/GI/1 system), it may be quite involved
to obtain these values. Abate and Whitt [6] discuss the solution of functional equations for complexarguments, and provide conditions for iterative methods to converge. Variants of these methods can
a.o. be found in [9] and [13]. Note finally that the Fourier-series method is closely related to the Laguerre
method (see e.g. [2, 3, 16]), i.e., the desired function is in both methods represented as an expansion in
terms of orthogonal functions, where the coefficients are expressed in terms of the transform.
A second class of approximate inversion techniques exists of analytical methods, which all more or less
follow a similar procedure. After determining the (asymptotic) tail behavior, one calculates the correspond-
ing parameters. Finally, approximate expressions for the tail probabilities are derived. The asymptotic
tail behavior of a probability distribution can be obtained analytically by calculating the value and type
of the rightmost singularity of the Laplace transform in the continuous case (see e.g. [8]), or by deter-
mining the value and type of the singularity with the smallest modulus of the pgf in the discrete case
(see e.g. [10]). Choudhury and Lucantoni [11] showed further that high-order moments of the stochastic
variable can be used to estimate the asymptotic parameters of the cdf. Abate et al [1] provide theoret-
ical support for this moment-based algorithm, and present new refined estimators which converge much
faster than the estimators proposed in [11]. The techniques in [1] were also used in [22] for computing
the asymptotic parameters numerically. When the transforms are not available explicitly, as in models of
busy-periods or polling systems, moment-based algorithms prove useful (see e.g. [12]). Finally, when the
transforms are only available in matrix-form, one can use an analytical method based on the dominant
eigenvalue of the transform-matrix and on the Chernoff large deviations approximation (see e.g. [15]).
In summary, in the analytical approach, an approximate expression is found in terms of a limited
number of parameters (such as the dominant singularity of the pgf), while the numerical approach only
yield ’a set of numbers’. The analytical method has the advantage that the behavior of the pmf is found. On
the other hand, numerical inversion techniques are usually more accurate for low arguments of the cdf’s
or pmf’s. The analytical approach thus complements the numerical one.
In this paper, we use the dominant singularity method for deriving approximate expressions for tail
probabilities of a discrete-time variable from its pgf. E.g. Bruneel et al [10] have shown that for high n, the
pmf x(n) of a discrete variable X is dominated by the contribution of the singularity of the corresponding
pgf, with the smallest absolute value. This dominant singularity is necessarily positive real and larger
than 1. In traditional single-class queueing systems with a FIFO scheduling discipline, the pgf’s of the
system quantities generally have one type of dominant singularity, usually a simple pole (i.e., a zero of
the denominator of the pgf with multiplicity 1) (see e.g. [10]). This leads to the well-known geometric (or
exponential) behavior x(n) ≈ Ks−n−1∗
, with s∗ the dominant pole of the corresponding pgf.
In e.g. priority queueing systems however, several (types of) singularities may play a role (see e.g. [8, 17,
26]). In [17] and [26], the authors analyse “basic” discrete-time priority queueing systems: two-class queues
with single-slot service times and with a HOL (Head-Of-the-Line) priority scheduling discipline. E.g. in[26], it is shown that two (types of) singularities on the positive real axis of the pgf of the low-priority
packet delay in such a priority system play a role: a simple pole and a branch point. This branch point
is a result of an implicitly defined function appearing in the pgf. Both singularities can dominate and
it depends on the system parameters (arrival rates in that case) which one is dominant. A consequence
of the appearance of two types of singularities is that two forms of tail behavior can distinguished be,
namely exponential behavior when the simple pole dominates and non-exponential tail behavior when
the branch point dominates. Tail behavior in continuous-time priority systems has been examined, via
analytical methods in e.g. [1, 8, 11, 22], or via numerical methods in e.g. [23]. We finally note that the
papers mentioned in this paragraph all assume infinite queue sizes. Different results are however obtained
by scaling the number of arrival sources along with the capacity of the system and the queue size (see
e.g. [20]).
In this paper, we calculate the tail probabilities of two particular random variables in more complex
discrete-time priority queueing systems, whereby more than two (types of) singularities may exist and
each of them may (co-)dominate, depending on the values of the various system parameters. We derive
Until now, we have focused on the potential poles of the denominator and on the branch point of
the implicitly defined function Y (z). Thereby, we have (implicitly) assumed that all pgf’s appearing in
expression (1) are analytic in the region of these singularities, i.e., that the radii of convergence of these
pgf’s do not play a role as possible dominant singularities of U T (z). In this subsection, we will prove that
this is indeed the case for all pgf’s appearing in (1), except for one, namely S 2(AT (z)). First, we will show
that the radii of convergence of S 1(AT (z)), AT (z), S 2(A(Y (z), z)), A(Y (z), z) are necessarily larger than
at least one of the singularities sB, sL or sT .
We first focus on the radius of convergence of S 1(AT (z)). We distinguish two cases: sT is a singularity
or sT is not a singularity (see subsection 2.3). In the first case, sT is within the region of convergence of
S 1(AT (z)), since sT > 1 is a solution of z − S 1(AT (z)) = 0. In the second case, one can prove that |z| <
|Y (z)| for |z| larger than the largest zero of z−S 1(AT (z)) (i.e., 1 or sT ). For those z, |Y (z)| > |S 1(AT (z))|and as a result, Y (z) will reach its branch point sB before S 1(AT (z)) diverges. Concluding, the radius of
convergence of S 1(AT (z)) is in both cases preceded by another singularity of U T (z).
Further, it is easily verified that the radius of convergence of S 2(A(Y (z), z)) is not a new potential
singularity. Again two cases are distinguished: sL exists or sL does not exist (see subsection 2.4). In the
first case, S 2(A(Y (z), z)) < z for 1 < z < sL sincedS 2(A(Y (z), z))
dz
z=1
< 1, S 2(A(Y (sL), sL)) = sL,
and S 2(A(Y (z), z)) is a convex function. As a result, the radius of convergence of S 2(A(Y (z), z)) is larger
than sL. In the second case, S 2(A(Y (z), z)) < z for 1 < z < sB sincedS 2(A(Y (z), z))
dz z=1
< 1 and Y (z)
reaches its branch point before S 2(A(Y (z), z)) reaches z. As a result, sB is the radius of convergence of
S 2(A(Y (z), z)), and we have already treated this singularity in subsection 2.2.
Thirdly, since the service times have means bigger than or equal to 1, S 1(AT (z)) ≥ AT (z) and
S 2(A(Y (z), z)) ≥ A(Y (z), z), for z larger than 1 and positive real. As a result, the radii of convergence of
AT (z) and A(Y (z), z) are larger (or equal) than the radii of convergence of S 1(AT (z)) and S 2(A(Y (z), z))
respectively, and thus do not play a role (a fortiori) in the tail behavior of U T (z) (or do not yield new
potential singularities).
Finally, as already mentioned, the radius of convergence of S 2(AT (z)), denoted by sQ, can be the
dominant singularity of U T (z). The reason why this singularity can be dominant is that S 2(z) does not
influence the singularities sB and sT , as can be seen from subsections 2.2 and 2.3 respectively. Hence,
S 2(z) can be such that S 2(AT (z)) →∞ before z reaches sB or sT . Furthermore, S 2(z) can be such that
S 2(AT (z)) > S 2(A(Y (z), z)) for z > 1, and even so that for z increasing, S 2(AT (z)) reaches its radius of
convergence sQ before U T (z) reaches sL. As a result, sQ can be smaller than sB, sT and sL, and thus be
the dominant singularity of U T (z). We will give an example of such S 2(z) in the following subsection.
Figure 5: Tail behavior of U T (z) as a function of ρ1 and ρ2, for geometric service times
two-dimensional binomial arrival process but for geometric service times with pgf’s S j(z) =z
z − µj(z − 1)( j = 1, 2), with µ1 = 2, µ2 = 4 (Figure 5a.) and µ1 = 4, µ2 = 2 (Figure 5b.). We observe that the (ρ1, ρ2)-
space again is split in several regions. sT , sL, or sQ determine the tail behavior of U T (z) when µ1 = 2,
µ2 = 4 (see Figure 5a.), while U T (z) is dominated by sT , sL, or sB when µ1 = 4, µ2 = 2 (see Figure
5b.). When the service times are geometrically distributed, the radius of convergence of S 2(AT (z)) may
thus determine the tail behavior of the total system contents.
2.7 The behavior of U T (z) in its dominant singularity
The type of the (co-)dominant singularity has a large impact on the tail behavior (see e.g. [8, 17]). In
this subsection, we use s∗ as a general notation for the dominant singularity of U T (z). According tothe previous subsection, three possible cases are established for s∗: s∗ = sB, s∗ = sB and sB is single-
dominant, or s∗ = sB but other singularities are co-dominant. In the remainder, we formulate a procedure
to approximate U T (z) in the neighbourhood of s∗, for each case. We refer to Appendix B for applications
of this procedure.
In the first case, the branch point sB is not dominant and a ’regular’ pole is the dominant singu-
larity. We replace each factor of U T (z) by its nth-order Taylor-series approximation in s∗, with n the
multiplicity of s∗ as zero of that factor. This leads to U T (z) ≈ K (∗)T
(s∗−
z)mfor z → s∗, with K
(∗)T a constant
and with m the multiplicity of the dominant singularity.
In the second case, in which the branch point sB is the only dominant singularity, we first substitute ex-
pression (3) of Y (z) in (1). Secondly, the obtained expression is rationalised, i.e., all roots are removed from
the denominator. We furthermore replace each factor of the denominator by the 0th-order Taylor-series
approximation in sB (since sB is not a zero of the denominator). We then get an expression for U T (z) of the
form U T (sB)−K (∗)T (sB − z)1/2 −K
(∗∗)T (sB − z) in the neighourhood of sB. Since the last term tends to
zero faster than the second term, we can omit the last term, yielding U T (z) ≈ U T (sB)−K (∗)T (sB − z)1/2
Figure 8: The functions z and AT (z) for z real and positive
0
2
4
6
8
10
0 0.5 1 1.5 2 2.5 3
zv1 v2
1
dT
V0(z)
AT(V0(z))
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5 3
zv1
V0(z)
AT(V0(z))
a. dV exists b. dV does not exist
Figure 9: The singularity dV
3.4 Singularity v2
Thirdly, we look at the zeros of V 0(z) − AT (V 0(z)), which may be singularities of (7), since V 0(z) −AT (V 0(z)) is a factor of the denominator. We first rewrite V 0(z)− AT (V 0(z)) as the following system of
equations:
x−AT (x) = 0
x = V 0(z). (11)
The equation x − AT (x) = 0 has two positive real solutions, namely x = 1 and x = dT (see subsection
3.3). So V 0(z) − AT (V 0(z)) may have two real positive solutions v1 and v2, satisfying V 0(v1) = 1 and
V 0(v2) = dT respectively (see Figure 9a.). However, v1 is never a singularity of D2(z) since the numerator
of (7) is also zero for V 0(v1) = 1. Secondly, v2 does not always exist, since V 0(z) ceases to exist for z > dB,
and thus the second solution is not always ’reached’ before dB (see Figure 9a.). Whether the singularity
v2 exists or not, depends on the values of all system parameters: the arrival process and the jumping
probability β . When v2 exists, v2 =dT
(1− β )A1(dT ), which is easily checked by substituting V 0(z) by dT
in the definition of V 0(z). So, in summary, three cases can occur for the potential singularity v2: v2 exists
and v2 < dB, v2 exists and v2 = dB or v2 does not exist.
3.5 Singularity d1
A fourth potential singularity of D2(z) on the real positive axis (> 1) - denoted by d1 - is given by
the zero of 1 − (1 − β )A1(z). d1 is however not always a singularity, as we will show in the remainder
of this subsection. It is easily seen that (x, z) = (d1, d1) is a solution of x − (1 − β )zA1(x) = 0. This
equation, which has been discussed in subsection 3.2, has no solutions for z > dB, positive real. Hence,
d1 has to be smaller than dB . For z < dB , this equation has two positive real solutions (x, z), namely
(V 0(z), z) and (V ∗0 (z), z). Consequently, d1 = V 0(d1) or d1 = V ∗0 (d1) (see Figure 10). We can now verify
that when d1 = V 0(d1), d1 is also a zero of the numerator of D2(z) (see expression (7)), and thus not asingularity of D2(z). On the other hand, when d1 = V ∗0 (d1), d1 is not a zero of the numerator, and is
thus a singularity of D2(z). To conclude this subsection, we state the three possible cases for the potential
singularity d1: d1 = V 0(d1) < V 0(dB), d1 = V 0(dB) (in which case the branch point dB and d1 coincide)
and d1 = V ∗0 (d1) > V 0(dB). d1 is a singularity in the second and third case.
3.6 Determining the tail probabilities
First note that it can be proven that in this case the radii of convergence of the generating functions
appearing in (7) are never dominant. We can thus bring everything together: the singularities dB, dT , v2
or d1 - depending on which one is dominant - characterize the tail behavior of the class-2 delay. Their
mutual behavior, illustrated in Figures 11a. and 11b. for a two-dimensional binomial arrival process, and
for β = 0.4 and β = 0.75 respectively, can be determined in a similar way as in subsection 2.6: first
we calculate for which combinations of class-1 and class-2 loads singularities coincide, and then for each
region we determine which singularity is dominant.
Remark that in the area above the linear line in Figures 11a. and 11b. (defined by λ1+λ2 = 1), the total